In quantum information processing tasks, we usually encode information in multipartite quantum states with high amount of quantum correlations between the different parties of the system [1]}, [2]}, [3]}, [4]}. In this sense, it is always needed to find the ways to enhance and to protect the degree of entanglement between the components of the quantum system employed to implement quantum protocols especially in presence of effects such as photon-substraction, photo addition, and their superposition, mixing local squeezing and local displacement. In this context, recently some useful tools to generate and to enhance the entanglement degree between two and three modes of quantized radiation field.
This is essentially due to the advent of experimental quantum optics of quantum optics allows to produce and manipulate various non-classical optical fields [5]}, [6]}. The photon addition and subtraction modeled mathematically by the actions of bosons creation operators \(a^\dag \)  and annihilation operators \(a\) , have prompted significant investigations [7]}, [8]}, [9]}. As quantum entanglement is a crucial part of quantum information theory and is at the heart of many quantum technologies, a lot of attention was dedicated to the generation of this resource using optical states to encode the quantum information to be processed to implement quantum protocols such as [10]}, [11]}, [12]}, quantum teleportation [13]}, [14]}, quantum dense coding [15]}, [16]}, quantum key distribution [17]}, quantum cryptography [15]}, [17]}, and many other applications. In exploiting quantum correlations in multi-particle quantum system, a classification of the states according their degree of entanglement was considered in several works. The states violating Bell inequalities were studied in [20]}, [21]}, [22]}, [23]}. The W-type and GHZ-type entangled states were considered in [24]}, [25]}.
